
They are a family of interacting 4d \(\mathcal {N}=2\)  SCFTs labelled by a positive integer \(N\) , which we call the rank of the \(N_f=4\)  theory. As SCFTs, they are invariant under the 4d \(\mathcal {N}=2\)  superconformal group \(\mathrm {SU}(2,2|2)\) .

The \(N_f=4\)  SCFTs have an exact \(\mathrm {SL}(2,\mathbb {Z})\)  S-duality. This means that each theory has a one-complex-dimensional conformal manifold given by the upper half complex plane modulo \(\mathrm {SL}(2,\mathbb {Z})\)  Möbius transformations. Though the center of \(\mathrm {SL}(2, \mathbb {Z})\)  acts trivially on the conformal manifold, it acts non-trivially as charge conjugation in the field theory. Around a special point on the conformal manifold the theory admits a weakly-coupled Lagrangian description in terms of \(\mathrm {USp}(2N)\)  gauge theory with 4 fundamental and 1 antisymmetric hypermultiplets. The weak coupling limit of the complex gauge coupling constant \({\tau }\)  parameterizing the conformal manifold is \(\mathrm {Im}({\tau }) \rightarrow \infty \) .

The internal global “flavor" symmetry is \(\mathrm {Spin}(8)\)  for \(N=1\)  and \(\mathrm {Spin}(8) \times \mathrm {SU}(2)\)  for \(N \ge 2\) , under which the four fundamental hypermultiplets (the same as eight fundamental half-hypermultiplets) transform in the \((8_v,1)\)  representation, and the antisymmetric hypermultiplet in the \((1, 2)\)  representation. Correspondingly, there is a space of \(\mathcal {N}=2\) -preserving mass deformations given by the complexified weight space of \(\mathrm {Spin}(8) \times \mathrm {SU}(2)\) . Introduce mass (or deformation) parameters \((m_1,\ldots ,m_4)\)  for \(\mathrm {Spin}(8)\)  and \(M\)  for \(\mathrm {SU}(2)\)  as linear coordinates on this parameter space such that \(m_i\)  is the complex mass of the \(i\) -th fundamental hypermultiplet, and \(M\)  the mass of the antisymmetric hypermultiplet.We use an unconventional normalization for the mass such that our masses \(m\)  are related to the canonically normalized masses \(\widetilde{m}\)  by \(\widetilde{m} = \sqrt{2} \, m\) . The principal congruence subgroup \({\Gamma }(2)\subset \mathrm {SL}(2,\mathbb {Z})\)  of the S-duality group acts trivially on the \(\mathrm {Spin}(8)\)  masses, while the quotient \(\mathrm {SL}(2, \mathbb {Z})/{\Gamma }(2) \simeq S_3\)  transforms the mass parameters by the \(\mathrm {Spin}(8)\)  “triality" outer automorphism [1]}, [2]}. The antisymmetric mass is invariant under S-duality transformations.

The operator content of an \(N_f=4\)  theory can be organized in terms of the unitary representations of its global symmetry \(\mathrm {SU}(2,2|2) \times \mathrm {Spin}(8) \times \mathrm {SU}(2)\) . In particular, with respect to \(\mathrm {SU}(2,2|2)\)  there are various sectors of supersymmetry-protected BPS operators, for instance, Coulomb branch operators and Higgs branch operators. The condensate of the scalar components in the \(\mathcal {N}=2\)  multiplets of BPS operators parameterize moduli spaces of \(\mathcal {N}=2\)  invariant vacuum states.

The moduli space of vacua consists of various branches each of which is locally a metric product of a Coulomb factor and a Higgs factor, with complex dimension \(n_C\)  and quaternionic dimension \(n_H\) , respectively. Conventionally, the branch with maximal \(n_C\)  is called the Coulomb branch and the branch with maximal \(n_H\)  the Higgs branch. The rank \(N\)  \(N_f=4\)  theory has a Coulomb branch with \((n_C, n_H) = (N, N-1)\)  and a Higgs branch with \((n_C, n_H) = (0, 6 N-1)\) . The \(N-1\)  quaternionic dimensional Higgs factor of the Coulomb branch comes from the components of the antisymmetric hypermultiplet carrying zero weight with respect to the \(\mathrm {USp}(2N)\)  gauge algebra.

The vector multiplet of the Lagrangian theory contains a scalar field \(\Phi \)  in the adjoint representation. The Coulomb branch coordinate ring is freely generated by \(u_i := \mathrm {tr}(\wedge ^{2i}\Phi )\)  with \(i = 1, 2, \ldots , N\) , corresponding to the primitive Casimir elements of \(\mathrm {USp}(2N)\) . The Coulomb branch coordinate ring is graded by the scaling dimension, so the weight of \(u_i\)  is \(2i\) . Since the Coulomb branch chiral operators are BPS operators, this description of the Coulomb branch chiral ring is true at all points of the conformal manifold, not just at the weak coupling point.

